The cold dry air can cause your blood vessels in your nose to crack and cause your nose to bleed which is normal and usually nothing to worry about.
You can help keep the nostrils moist and stop the nosebleed by putting some petroleum jelly on cotton balls and then putting the cotton balls safely into your nostrils which should stop the nosebleed eventually.
If your nose is still bleeding then you should see a doctor about it but usually the nosebleeds will stop on their own.